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## G = D28⋊5S3order 336 = 24·3·7

### The semidirect product of D28 and S3 acting through Inn(D28)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — D28⋊5S3
 Chief series C1 — C7 — C21 — C42 — C6×D7 — Dic3×D7 — D28⋊5S3
 Lower central C21 — C42 — D28⋊5S3
 Upper central C1 — C2 — C4

Generators and relations for D285S3
G = < a,b,c,d | a28=b2=c3=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a14b, dcd=c-1 >

Subgroups: 420 in 80 conjugacy classes, 32 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C6, C7, C2×C4, D4, Q8, Dic3, Dic3, C12, D6, C2×C6, D7, C14, C14, C4○D4, C21, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, Dic7, C28, C28, D14, C2×C14, S3×C7, C3×D7, C42, D42S3, Dic14, C4×D7, D28, C7⋊D4, C2×C28, C7×Dic3, Dic21, C84, C6×D7, S3×C14, C4○D28, Dic3×D7, C21⋊D4, C3×D28, S3×C28, Dic42, D285S3
Quotients: C1, C2, C22, S3, C23, D6, D7, C4○D4, C22×S3, D14, D42S3, C22×D7, S3×D7, C4○D28, C2×S3×D7, D285S3

Smallest permutation representation of D285S3
On 168 points
Generators in S168
```(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28)(29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168)
(1 7)(2 6)(3 5)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)(29 53)(30 52)(31 51)(32 50)(33 49)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)(54 56)(57 83)(58 82)(59 81)(60 80)(61 79)(62 78)(63 77)(64 76)(65 75)(66 74)(67 73)(68 72)(69 71)(85 105)(86 104)(87 103)(88 102)(89 101)(90 100)(91 99)(92 98)(93 97)(94 96)(106 112)(107 111)(108 110)(113 131)(114 130)(115 129)(116 128)(117 127)(118 126)(119 125)(120 124)(121 123)(132 140)(133 139)(134 138)(135 137)(141 157)(142 156)(143 155)(144 154)(145 153)(146 152)(147 151)(148 150)(158 168)(159 167)(160 166)(161 165)(162 164)
(1 81 106)(2 82 107)(3 83 108)(4 84 109)(5 57 110)(6 58 111)(7 59 112)(8 60 85)(9 61 86)(10 62 87)(11 63 88)(12 64 89)(13 65 90)(14 66 91)(15 67 92)(16 68 93)(17 69 94)(18 70 95)(19 71 96)(20 72 97)(21 73 98)(22 74 99)(23 75 100)(24 76 101)(25 77 102)(26 78 103)(27 79 104)(28 80 105)(29 165 138)(30 166 139)(31 167 140)(32 168 113)(33 141 114)(34 142 115)(35 143 116)(36 144 117)(37 145 118)(38 146 119)(39 147 120)(40 148 121)(41 149 122)(42 150 123)(43 151 124)(44 152 125)(45 153 126)(46 154 127)(47 155 128)(48 156 129)(49 157 130)(50 158 131)(51 159 132)(52 160 133)(53 161 134)(54 162 135)(55 163 136)(56 164 137)
(1 45)(2 46)(3 47)(4 48)(5 49)(6 50)(7 51)(8 52)(9 53)(10 54)(11 55)(12 56)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)(28 44)(57 130)(58 131)(59 132)(60 133)(61 134)(62 135)(63 136)(64 137)(65 138)(66 139)(67 140)(68 113)(69 114)(70 115)(71 116)(72 117)(73 118)(74 119)(75 120)(76 121)(77 122)(78 123)(79 124)(80 125)(81 126)(82 127)(83 128)(84 129)(85 160)(86 161)(87 162)(88 163)(89 164)(90 165)(91 166)(92 167)(93 168)(94 141)(95 142)(96 143)(97 144)(98 145)(99 146)(100 147)(101 148)(102 149)(103 150)(104 151)(105 152)(106 153)(107 154)(108 155)(109 156)(110 157)(111 158)(112 159)```

`G:=sub<Sym(168)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,53)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(54,56)(57,83)(58,82)(59,81)(60,80)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(85,105)(86,104)(87,103)(88,102)(89,101)(90,100)(91,99)(92,98)(93,97)(94,96)(106,112)(107,111)(108,110)(113,131)(114,130)(115,129)(116,128)(117,127)(118,126)(119,125)(120,124)(121,123)(132,140)(133,139)(134,138)(135,137)(141,157)(142,156)(143,155)(144,154)(145,153)(146,152)(147,151)(148,150)(158,168)(159,167)(160,166)(161,165)(162,164), (1,81,106)(2,82,107)(3,83,108)(4,84,109)(5,57,110)(6,58,111)(7,59,112)(8,60,85)(9,61,86)(10,62,87)(11,63,88)(12,64,89)(13,65,90)(14,66,91)(15,67,92)(16,68,93)(17,69,94)(18,70,95)(19,71,96)(20,72,97)(21,73,98)(22,74,99)(23,75,100)(24,76,101)(25,77,102)(26,78,103)(27,79,104)(28,80,105)(29,165,138)(30,166,139)(31,167,140)(32,168,113)(33,141,114)(34,142,115)(35,143,116)(36,144,117)(37,145,118)(38,146,119)(39,147,120)(40,148,121)(41,149,122)(42,150,123)(43,151,124)(44,152,125)(45,153,126)(46,154,127)(47,155,128)(48,156,129)(49,157,130)(50,158,131)(51,159,132)(52,160,133)(53,161,134)(54,162,135)(55,163,136)(56,164,137), (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,43)(28,44)(57,130)(58,131)(59,132)(60,133)(61,134)(62,135)(63,136)(64,137)(65,138)(66,139)(67,140)(68,113)(69,114)(70,115)(71,116)(72,117)(73,118)(74,119)(75,120)(76,121)(77,122)(78,123)(79,124)(80,125)(81,126)(82,127)(83,128)(84,129)(85,160)(86,161)(87,162)(88,163)(89,164)(90,165)(91,166)(92,167)(93,168)(94,141)(95,142)(96,143)(97,144)(98,145)(99,146)(100,147)(101,148)(102,149)(103,150)(104,151)(105,152)(106,153)(107,154)(108,155)(109,156)(110,157)(111,158)(112,159)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,53)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(54,56)(57,83)(58,82)(59,81)(60,80)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(85,105)(86,104)(87,103)(88,102)(89,101)(90,100)(91,99)(92,98)(93,97)(94,96)(106,112)(107,111)(108,110)(113,131)(114,130)(115,129)(116,128)(117,127)(118,126)(119,125)(120,124)(121,123)(132,140)(133,139)(134,138)(135,137)(141,157)(142,156)(143,155)(144,154)(145,153)(146,152)(147,151)(148,150)(158,168)(159,167)(160,166)(161,165)(162,164), (1,81,106)(2,82,107)(3,83,108)(4,84,109)(5,57,110)(6,58,111)(7,59,112)(8,60,85)(9,61,86)(10,62,87)(11,63,88)(12,64,89)(13,65,90)(14,66,91)(15,67,92)(16,68,93)(17,69,94)(18,70,95)(19,71,96)(20,72,97)(21,73,98)(22,74,99)(23,75,100)(24,76,101)(25,77,102)(26,78,103)(27,79,104)(28,80,105)(29,165,138)(30,166,139)(31,167,140)(32,168,113)(33,141,114)(34,142,115)(35,143,116)(36,144,117)(37,145,118)(38,146,119)(39,147,120)(40,148,121)(41,149,122)(42,150,123)(43,151,124)(44,152,125)(45,153,126)(46,154,127)(47,155,128)(48,156,129)(49,157,130)(50,158,131)(51,159,132)(52,160,133)(53,161,134)(54,162,135)(55,163,136)(56,164,137), (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,43)(28,44)(57,130)(58,131)(59,132)(60,133)(61,134)(62,135)(63,136)(64,137)(65,138)(66,139)(67,140)(68,113)(69,114)(70,115)(71,116)(72,117)(73,118)(74,119)(75,120)(76,121)(77,122)(78,123)(79,124)(80,125)(81,126)(82,127)(83,128)(84,129)(85,160)(86,161)(87,162)(88,163)(89,164)(90,165)(91,166)(92,167)(93,168)(94,141)(95,142)(96,143)(97,144)(98,145)(99,146)(100,147)(101,148)(102,149)(103,150)(104,151)(105,152)(106,153)(107,154)(108,155)(109,156)(110,157)(111,158)(112,159) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28),(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)], [(1,7),(2,6),(3,5),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,20),(17,19),(29,53),(30,52),(31,51),(32,50),(33,49),(34,48),(35,47),(36,46),(37,45),(38,44),(39,43),(40,42),(54,56),(57,83),(58,82),(59,81),(60,80),(61,79),(62,78),(63,77),(64,76),(65,75),(66,74),(67,73),(68,72),(69,71),(85,105),(86,104),(87,103),(88,102),(89,101),(90,100),(91,99),(92,98),(93,97),(94,96),(106,112),(107,111),(108,110),(113,131),(114,130),(115,129),(116,128),(117,127),(118,126),(119,125),(120,124),(121,123),(132,140),(133,139),(134,138),(135,137),(141,157),(142,156),(143,155),(144,154),(145,153),(146,152),(147,151),(148,150),(158,168),(159,167),(160,166),(161,165),(162,164)], [(1,81,106),(2,82,107),(3,83,108),(4,84,109),(5,57,110),(6,58,111),(7,59,112),(8,60,85),(9,61,86),(10,62,87),(11,63,88),(12,64,89),(13,65,90),(14,66,91),(15,67,92),(16,68,93),(17,69,94),(18,70,95),(19,71,96),(20,72,97),(21,73,98),(22,74,99),(23,75,100),(24,76,101),(25,77,102),(26,78,103),(27,79,104),(28,80,105),(29,165,138),(30,166,139),(31,167,140),(32,168,113),(33,141,114),(34,142,115),(35,143,116),(36,144,117),(37,145,118),(38,146,119),(39,147,120),(40,148,121),(41,149,122),(42,150,123),(43,151,124),(44,152,125),(45,153,126),(46,154,127),(47,155,128),(48,156,129),(49,157,130),(50,158,131),(51,159,132),(52,160,133),(53,161,134),(54,162,135),(55,163,136),(56,164,137)], [(1,45),(2,46),(3,47),(4,48),(5,49),(6,50),(7,51),(8,52),(9,53),(10,54),(11,55),(12,56),(13,29),(14,30),(15,31),(16,32),(17,33),(18,34),(19,35),(20,36),(21,37),(22,38),(23,39),(24,40),(25,41),(26,42),(27,43),(28,44),(57,130),(58,131),(59,132),(60,133),(61,134),(62,135),(63,136),(64,137),(65,138),(66,139),(67,140),(68,113),(69,114),(70,115),(71,116),(72,117),(73,118),(74,119),(75,120),(76,121),(77,122),(78,123),(79,124),(80,125),(81,126),(82,127),(83,128),(84,129),(85,160),(86,161),(87,162),(88,163),(89,164),(90,165),(91,166),(92,167),(93,168),(94,141),(95,142),(96,143),(97,144),(98,145),(99,146),(100,147),(101,148),(102,149),(103,150),(104,151),(105,152),(106,153),(107,154),(108,155),(109,156),(110,157),(111,158),(112,159)]])`

51 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 6A 6B 6C 7A 7B 7C 12 14A 14B 14C 14D ··· 14I 21A 21B 21C 28A ··· 28F 28G ··· 28L 42A 42B 42C 84A ··· 84F order 1 2 2 2 2 3 4 4 4 4 4 6 6 6 7 7 7 12 14 14 14 14 ··· 14 21 21 21 28 ··· 28 28 ··· 28 42 42 42 84 ··· 84 size 1 1 6 14 14 2 2 3 3 42 42 2 28 28 2 2 2 4 2 2 2 6 ··· 6 4 4 4 2 ··· 2 6 ··· 6 4 4 4 4 ··· 4

51 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 S3 D6 D6 D7 C4○D4 D14 D14 D14 C4○D28 D4⋊2S3 S3×D7 C2×S3×D7 D28⋊5S3 kernel D28⋊5S3 Dic3×D7 C21⋊D4 C3×D28 S3×C28 Dic42 D28 C28 D14 C4×S3 C21 Dic3 C12 D6 C3 C7 C4 C2 C1 # reps 1 2 2 1 1 1 1 1 2 3 2 3 3 3 12 1 3 3 6

Matrix representation of D285S3 in GL6(𝔽337)

 178 196 0 0 0 0 12 159 0 0 0 0 0 0 33 227 0 0 0 0 110 1 0 0 0 0 0 0 336 0 0 0 0 0 0 336
,
 336 0 0 0 0 0 160 1 0 0 0 0 0 0 304 110 0 0 0 0 33 33 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 336 0 0 0 0 1 336
,
 58 26 0 0 0 0 91 279 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 329 214 0 0 0 0 206 8

`G:=sub<GL(6,GF(337))| [178,12,0,0,0,0,196,159,0,0,0,0,0,0,33,110,0,0,0,0,227,1,0,0,0,0,0,0,336,0,0,0,0,0,0,336],[336,160,0,0,0,0,0,1,0,0,0,0,0,0,304,33,0,0,0,0,110,33,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,336,336],[58,91,0,0,0,0,26,279,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,329,206,0,0,0,0,214,8] >;`

D285S3 in GAP, Magma, Sage, TeX

`D_{28}\rtimes_5S_3`
`% in TeX`

`G:=Group("D28:5S3");`
`// GroupNames label`

`G:=SmallGroup(336,138);`
`// by ID`

`G=gap.SmallGroup(336,138);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-7,55,218,50,490,10373]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^28=b^2=c^3=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^14*b,d*c*d=c^-1>;`
`// generators/relations`

׿
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